The expected number of zeros of a random system of $p$-adic polynomials
We study the simultaneous zeros of a random family of $d$ polynomials in $d$ variables over the $p$-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the $d$-fold Cartesian product of the $p$-adic integers. Considering models in...
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| Main Author | |
|---|---|
| Format | Journal Article |
| Language | English |
| Published |
21.02.2006
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.math/0602478 |
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| Abstract | We study the simultaneous zeros of a random family of $d$ polynomials in $d$
variables over the $p$-adic numbers. For a family of natural models, we obtain
an explicit constant for the expected number of zeros that lie in the $d$-fold
Cartesian product of the $p$-adic integers. Considering models in which the
maximum degree that each variable appears is $N$, this expected value is \[
p^{d \lfloor \log_p N \rfloor} (1 + p^{-1} + p^{-2} + ... + p^{-d})^{-1} \] for
the simplest such model. |
|---|---|
| AbstractList | We study the simultaneous zeros of a random family of $d$ polynomials in $d$
variables over the $p$-adic numbers. For a family of natural models, we obtain
an explicit constant for the expected number of zeros that lie in the $d$-fold
Cartesian product of the $p$-adic integers. Considering models in which the
maximum degree that each variable appears is $N$, this expected value is \[
p^{d \lfloor \log_p N \rfloor} (1 + p^{-1} + p^{-2} + ... + p^{-d})^{-1} \] for
the simplest such model. |
| Author | Evans, Steven N |
| Author_xml | – sequence: 1 givenname: Steven N surname: Evans fullname: Evans, Steven N |
| BackLink | https://doi.org/10.48550/arXiv.math/0602478$$DView paper in arXiv |
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| Snippet | We study the simultaneous zeros of a random family of $d$ polynomials in $d$
variables over the $p$-adic numbers. For a family of natural models, we obtain
an... |
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| SubjectTerms | Mathematics - Commutative Algebra Mathematics - Probability |
| Title | The expected number of zeros of a random system of $p$-adic polynomials |
| URI | https://arxiv.org/abs/math/0602478 |
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